8.5 A variety of issues and questions

  1. A particularly interesting issue raised by our equations is that of the run-away (unstable) behavior of the equations of motion for a charged particle (with or without an external field). We saw in Equation (328View Equation) that there was a driving term in the equation of motion depending on the electric dipole moment (or the real center of charge). This driving term was totally independent of the real center of mass and thus does not lead to the classical instability. However, if we restrict the complex center of charge to be the same as the complex center of mass (a severe, but very attractive restriction leading to g = 2), then the innocuous driving dipole term becomes the classical radiation reaction term – suggesting instability. (Note that in this coinciding case there was no model building – aside from the coinciding lines – and no mass renormalization.)

    A natural question then is: does this unstable behavior really remain? In other words, is it possible that the large number of extra terms in the gravitational radiation reaction or the momentum recoil force might stabilize the situation? Answering this question is extremely difficult. If the gravitational effects do not stabilize, then – at least in this special case – the Einstein–Maxwell equations are unstable, since the run-away behavior would lead to an infinite amount of electromagnetic dipole energy loss.

    An alternative possible resolution to the classical run-away problem is simply to say that the classical electrodynamic model is wrong; and that one must treat the center of charge as different from the center of mass with its own dynamics.

  2. The interpretation and analysis of the complex analytic curves associated with the shear-free and asymptotically shear free null geodesic congruences naturally takes place in ℋ-space. Though we get extraordinarily attractive physical results – almost all coinciding with standard physical understanding – it nevertheless is a total mystery as to what it means or what is the physical significance of this complex Ricci-flat four-dimensional space. UpdateJump To The Next Update Information
  3. In our approximations, it was assumed that the complex world line yielded cuts of + ℑ that were close to Bondi cuts. At the present we do not have any straightforward means of finding the world lines and their associated cuts of ℑ+ that are far from the Bondi cuts.
  4. As mentioned earlier, when the gravitational and electromagnetic world lines coincide we find the rather surprising result of the Dirac value for the gyromagnetic ratio. Unfortunately, though this appears to be a significant result, we do not have any deeper understanding of it. It was simply there for us to observe.
  5. Is it possible that the complex structures that we have been seeing and using are more than just a technical device to organize ideas, and that they have a deeper significance? One direction to explore this is via Penrose’s twistor and asymptotic twistor theory. It is known that much of the material described here is closely related to twistor theory; an example is the fact that asymptotic shear-free NGCs are really a special case of the Kerr theorem, an important application of twistor theory (see Appendix A). This connection is being further explored.
  6. With much of the kinematics and dynamics of ordinary classical mechanics sitting in our results, i.e., in classical GR, is it possible that ordinary particle quantization could play a role in understanding quantum gravity? Attempts along this line have been made [147] but, so far, without much success.
  7. We reiterate that, a priori, there is no reason to suspect or believe that the world lines associated with shear-free congruences would allow the choice of a special congruence – and a special world line – to be singled out – and that furthermore it would be so connected with physical kinematics and dynamical laws. These results certainly greatly surprised and pleased us.
  8. As a final remark, we want to point out that there is an issue that we have ignored, namely, do the asymptotic solutions of the Einstein equations that we have discussed and used throughout this work really exist. By ‘really exist’ we mean the following: Are there, in sufficiently general circumstances, Cauchy surfaces with physically-given data, such that their evolution yields these asymptotic solutions? We have tacitly assumed throughout, with physical justification but no rigorous mathematical justification, that the full interior vacuum Einstein equations do lead to these asymptotic situations. However, there has been a great deal of deep and difficult analysis [131011] showing, in fact, that large classes of solutions to the Cauchy problem with physically-relevant data do lead to the asymptotic behavior we have discussed. Recently there has been progress made on the same problem for the Einstein–Maxwell equations.
  9. An interesting issue, not yet explored but potentially important, is what can be said about the structure of ℋ-space where there are special points that are related to the real cuts of null infinity.UpdateJump To The Next Update Information


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